Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > xnn0xrnemnf | GIF version |
Description: The extended nonnegative integers are extended reals without negative infinity. (Contributed by AV, 10-Dec-2020.) |
Ref | Expression |
---|---|
xnn0xrnemnf | ⊢ (𝐴 ∈ ℕ0* → (𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xnn0xr 9045 | . 2 ⊢ (𝐴 ∈ ℕ0* → 𝐴 ∈ ℝ*) | |
2 | xnn0nemnf 9051 | . 2 ⊢ (𝐴 ∈ ℕ0* → 𝐴 ≠ -∞) | |
3 | 1, 2 | jca 304 | 1 ⊢ (𝐴 ∈ ℕ0* → (𝐴 ∈ ℝ* ∧ 𝐴 ≠ -∞)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1480 ≠ wne 2308 -∞cmnf 7798 ℝ*cxr 7799 ℕ0*cxnn0 9040 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1re 7714 ax-addrcl 7717 ax-rnegex 7729 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-uni 3737 df-int 3772 df-pnf 7802 df-mnf 7803 df-xr 7804 df-inn 8721 df-n0 8978 df-xnn0 9041 |
This theorem is referenced by: xnn0xadd0 9650 xnn0add4d 9669 |
Copyright terms: Public domain | W3C validator |